# What is a hypercomplex quantity?

An extract from the book "Quantum Mechanics and the Path Integrals" by Richard P. Feynman and A. R. Hibbs:

We can state the correct law for $$P(x)$$ mathematically by saying that $$P(x)$$ is the absolute square of a certain complex quantity (if electron spin is taken into account, it is a hypercomplex quantity) $$\phi(x)$$ which we call the probability amplitude of arrival at $$x$$.

What does hypercomplex quantity mean here?

## 1 Answer

Usually "hypercomplex" means a quaternion, but he just means that the wavefunction at a point $$x$$ has two components $$\psi_\uparrow(x)$$ and $$\psi_\downarrow(x)$$ with the probability density of being found at $$x$$ being $$P(x)= |\psi_\uparrow(x)|^2+ |\psi_\downarrow(x)|^2.$$

• Thanks for the answer mate! Can you give good ref. to read more about Quaternions? – Vidya Sagar V Sep 30 '18 at 14:07
• Thw wikipedia article "Quaternion" seems quite comprehensive. – mike stone Oct 1 '18 at 17:51
• I found this video by 3blue1brown helpful. – Vidya Sagar V Oct 15 '18 at 10:18